Formal Realization of AxE in Reactive Systems (LaTeX)

(( Version 5.04.2025 ))

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Operational Realization of Axiomatic Erosion in Reactive Systems

(A Coalgebraic Proof via the Differentiation–Binding Functor)

We model open reactive systems as coalgebras of the composite endofunctor
F = Pf ◦∆ : Set → Set, where ∆ collects unordered event-pairs (excendence S(e)) and Pf forms coherent histories (incendence S(i)). This composition defines the substrative frequency S(∞) = S(i) ⊗ S(e), whose terminal coalgebra (Ω, ζ) realizes the universal recursive process central to Breeze Theory. We prove that every finite labelled-transition system (LTS) canonically embeds into Ω, and that any undecidable predicate on Ω remains undecidable on some finite subsystem. This operationalizes the principle of Axiomatic Erosion — that self-referential systems inevitably generate undecidable internal truths — and aligns Breeze Theory’s metaphysical core with established categorical semantics. The result: a fully formal, numerics-free foundation for structural recursion in computation, cognition, and physics.